A total-chromatic number analogue of plantholt's theorem

AbstractThe total chromatic number XT(G) of a graph G is the least number of colours needed to colour the edges and vertices of G so that no two adjacent vertices receive the same colour, no two edges incident with the same vertex receive the same colour, and no edge receives the same colour as either of the vertices it is incident with.Let n⩾1, let J be a subgraph of K2n, let e=∥E(J)∥ and let j(J) be the maximum size of a matching in J. Then XT(K2n[bsol]E(J))=2n+1 if and only if e+j≤n−1